Implementing Basic Arithmetic in $\mathbb{F}_p$ via $\mathbb{F}_2$, and Its Application for Computing the Hamming Distance of Linear Codes

2026-03-31

AI Summary

This work addresses the inefficiency of basic arithmetic operations over odd prime fields $\mathbb{F}_p$ and the high computational cost of determining the minimum Hamming distance of linear codes. To overcome these challenges, the authors propose an efficient arithmetic framework based on the binary field $\mathbb{F}_2$, leveraging pure bitwise operations to emulate $\mathbb{F}_p$ computations. Novel techniques such as isometric addition are introduced to accelerate distance calculations. Implemented in highly optimized C code, the approach demonstrates substantial performance gains over state-of-the-art systems like Magma and GAP/Guava across single-core, multi-core, and shared-memory multiprocessor architectures, offering a scalable high-performance solution for large-scale coding theory analysis.

Abstract

We present a new general method for performing basic arithmetic in the finite field~$\mathbb{F}_p$ for any prime $p>2$ by using traditional binary operations over~$\mathbb{F}_2$. Our new approach is efficient and competitive with current state-of-art methods. We apply our new arithmetic method to the computation of the minimum Hamming distance of random linear codes for the fields $\mathbb{F}_3$ and $\mathbb{F}_7$. Our new arithmetic method allows to apply new techniques such as the isometric addition that accelerate the computation of the Hamming distance. We have developed implementations in the C programming language for computing the Hamming distance that clearly outperform both state-of-art licensed software and open-source software such as \textsc{Magma} and \textsc{GAP}/\textsc{Guava} on single-core processors, multicore processors, and shared-memory multiprocessors.

Problem

• finite field arithmetic
• Hamming distance
• linear codes
• binary operations
• computational efficiency

Innovation

• finite field arithmetic
• binary representation
• Hamming distance
• isometric addition
• linear codes